tolkien - How well-known is the theology of Middle-earth, in Middle-earth?

Tolkien deliberately omitted references to "religion" in his works; see Letter 142:

I have not put in, or have cut out, practically all references to anything like 'religion', to cults or practices, in the imaginary world.

Letter 153, which he wrote to a reader who one must feel took many things entirely too seriously, elaborates further, and it's worth quoting the entirety of a footnote to it, although I'll break it up to avoid a "wall of text".

First a general overview:

There are thus no temples or 'churches' or fanes in this 'world' among 'good' peoples. They had little or no 'religion' in the sense of worship. For help they may call on a Vala (as Elbereth), as a Catholic might on a Saint, though no doubt knowing in theory as well as he that the power of the Vala was limited and derivative. But this is a 'primitive age': and these folk may be said to view the Valar as children view their parents or immediate adult superiors, and though they know they are subjects of the King he does not live in their country nor have there any dwelling.

On Hobbits:

I do not think Hobbits practised any form of worship or prayer (unless through exceptional contact with Elves).

Numenor, both before and after Sauron's arrival, including other Men who didn't go there:

The Númenóreans (and others of that branch of Humanity, that fought against Morgoth, even if they elected to remain in Middle-earth and did not go to Númenor: such as the Rohirrim) were pure monotheists. But there was no temple in Númenor (until Sauron introduced the cult of Morgoth). The top of the Mountain, the Meneltarma or Pillar of Heaven, was dedicated to Eru, the One, and there at any time privately, and at certain times publicly, God was invoked, praised, and adored: an imitation of the Valar and the Mountain of Aman. But Numenor fell and was destroyed and the Mountain engulfed, and there was no substitute.

And the Dunedain in Middle-earth:

Among the exiles, remnants of the Faithful who had not adopted the false religion nor taken part in the rebellion, religion as divine worship (though perhaps not as philosophy and metaphysics) seems to have played a small part; though a glimpse of it is caught in Faramir's remark on 'grace at meat'.

Letter 156 covers High Elves:

The High Elves were exiles from the Blessed Realm of the Gods (after their own particular
Elvish fall) and they had no 'religion' (or religious practices, rather) for those had been in the hands of the gods, praising and adoring Eru 'the One', Ilúvatar the Father of All on the Mt. of Aman.

There are many other mentions of religion in Middle-earth scattered throughout the Letters, but the general outcome and completely consistent observation is that Middle-earth has no religion as we would understand it today.